University of Hawaii ICS141: Discrete Mathematics for ...janst/141/lecture/07-Proofs.pdfDefinition: An integer n is called odd iff n=2k+1 for some integer k; n is even iff n=2k for

7-1 ICS 141: Discrete Mathematics I – Fall 2011

University of Hawaii

ICS141: Discrete Mathematics for Computer Science I

Dept. Information & Computer Sci., University of Hawaii

Jan Stelovsky based on slides by Dr. Baek and Dr. Still

Originals by Dr. M. P. Frank and Dr. J.L. Gross Provided by McGraw-Hill



Lecture 7

Chapter 1. The Foundations 1.6 Introduction to Proofs

Chapter 2. Basic Structures 2.1 Sets


University of Hawaii Proof Terminology n  A proof is a valid argument that establishes

the truth of a mathematical statement n  Axiom (or postulate): a statement that is

assumed to be true n  Theorem

n  A statement that has been proven to be true n  Hypothesis, premise

n  An assumption (often unproven) defining the structures about which we are reasoning


University of Hawaii More Proof Terminology n  Lemma

n  A minor theorem used as a stepping-stone to proving a major theorem.

n  Corollary n  A minor theorem proved as an easy

consequence of a major theorem. n  Conjecture

n  A statement whose truth value has not been proven. (A conjecture may be widely believed to be true, regardless.)


University of Hawaii Proof Methods n  For proving a statement p alone

n  Proof by Contradiction (indirect proof): Assume ¬p, and prove ¬p → F.


University of Hawaii Proof Methods n  For proving implications p → q, we have:

n  Trivial proof: Prove q by itself. n  Direct proof: Assume p is true, and prove q. n  Indirect proof:

n  Proof by Contraposition (¬q → ¬p): Assume ¬q, and prove ¬p.

n  Proof by Contradiction: Assume p ∧ ¬q, and show this leads to a contradiction. (i.e. prove (p ∧ ¬q) → F)

n  Vacuous proof: Prove ¬p by itself.


University of Hawaii Direct Proof Example n  Definition: An integer n is called odd iff n=2k+1

for some integer k; n is even iff n=2k for some k. n  Theorem: Every integer is either odd or even, but

not both. n  This can be proven from even simpler axioms.

n  Theorem: (For all integers n) If n is odd, then n2 is odd.

Proof: If n is odd, then n = 2k + 1 for some integer k.

Thus, n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. Therefore n2 is of the form 2j + 1 (with j the integer 2k2 + 2k), thus n2 is odd. ■


University of Hawaii Indirect Proof Example: Proof by Contraposition

n  Theorem: (For all integers n) If 3n + 2 is odd, then n is odd.

n  Proof: (Contrapositive: If n is even, then 3n + 2 is even) Suppose that the conclusion is false, i.e., that n is even. Then n = 2k for some integer k. Then 3n + 2 = 3(2k) + 2 = 6k + 2 = 2(3k + 1). Thus 3n + 2 is even, because it equals 2j for an integer j = 3k + 1. So 3n + 2 is not odd. We have shown that ¬(n is odd) → ¬(3n + 2 is odd), thus its contrapositive (3n + 2 is odd) → (n is odd) is also true. ■


University of Hawaii Vacuous Proof Example

n  Show ¬p (i.e. p is false) to prove p → q is true.

n  Theorem: (For all n) If n is both odd and even, then n2 = n + n.

n  Proof: The statement “n is both odd and even” is necessarily false, since no number can be both odd and even. So, the theorem is vacuously true. ■


University of Hawaii Trivial Proof Example n  Show q (i.e. q is true) to prove p → q is true.

n  Theorem: (For integers n) If n is the sum of two prime numbers, then either n is odd or n is even.

n  Proof: Any integer n is either odd or even. So the conclusion of the implication is true regardless of the truth of the hypothesis. Thus the implication is true trivially. ■


University of Hawaii Proof by Contradiction

n  A method for proving p. n  Assume ¬p, and prove both q and ¬q for

some proposition q. (Can be anything!) n  Thus ¬p → (q ∧ ¬q) n  (q ∧ ¬q) is a trivial contradiction, equal to F n  Thus ¬p → F, which is only true if ¬p = F n  Thus p is true


University of Hawaii Rational Number

n  Definition: The real number r is rational if there exist integers p and q with q ≠ 0 such that r = p/q. A real number that is not rational is called irrational.


University of Hawaii Proof by Contradiction Example

n  Theorem: is irrational. n  Proof: n  Assume that is rational. This means there are

integers x and y (y ≠ 0) with no common divisors such that = x/y. Squaring both sides, 2 = x2/y2, so 2y2 = x2. So x2 is even; thus x is even (see earlier). Let x = 2k. So 2y2 = (2k)2 = 4k2. Dividing both sides by 2, y2 = 2k2. Thus y2 is even, so y is even. But then x and y have a common divisor, namely 2, so we have a contradiction. Therefore, is irrational. ■

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University of Hawaii Proof by Contradiction n  Proving implication p → q by contradiction

n  Assume ¬q, and use the premise p to arrive at a contradiction, i.e. (¬q ∧ p) → F (p → q ≡ (¬q ∧ p) → F)

n  How does this relate to the proof by contraposition?

n  Proof by Contraposition (¬q → ¬p): Assume ¬q, and prove ¬p.


University of Hawaii Proof by Contradiction Example: Implication

n  Theorem: (For all integers n) If 3n + 2 is odd, then n is odd.

n  Proof: Assume that the conclusion is false, i.e., that n is even, and that 3n + 2 is odd. Then n = 2k for some integer k and 3n + 2 = 3(2k) + 2 = 6k + 2 = 2(3k + 1). Thus 3n + 2 is even, because it equals 2j for an integer j = 3k + 1. This contradicts the assumption “3n + 2 is odd”.

This completes the proof by contradiction, proving that if 3n + 2 is odd, then n is odd. ■


University of Hawaii Circular Reasoning n  The fallacy of (explicitly or implicitly) assuming

the very statement you are trying to prove in the course of its proof. Example:

n  Prove that an integer n is even, if n2 is even. n  Attempted proof:

Assume n2 is even. Then n2 = 2k for some integer k. Dividing both sides by n gives n = (2k)/n = 2(k/n). So there is an integer j (namely k/n) such that n = 2j. Therefore n is even. n  Circular reasoning is used in this proof.

Where? Begs the question: How do you show that j = k/n = n/2 is an integer,

without first assuming that n is even?



Chapter 2

Basic Structures: Sets, Functions, Sequences, and Sums


University of Hawaii 2.1 Sets

n  A set is a new type of structure, representing an unordered collection (group) of zero or more distinct (different) objects. The objects are called elements or members of the set.

n  Notation: x ∈ S

n  Set theory deals with operations between, relations among, and statements about sets.

n  Sets are ubiquitous in computer software systems. n  (E.g. data types Set, HashSet in java.util)


University of Hawaii Basic Notations for Sets n  For sets, we’ll use variables S, T, U,… n  We can denote a set S in writing by listing all of

its elements in curly braces: n  {a,b,c} is the set whose elements are a, b, and c

n  Set builder notation: n  For any statement P(x) over any domain,

{x | P(x)} is the set of all x such that P(x) is true

n  Example: {1, 2, 3, 4} = {x | x is an integer where x > 0 and x < 5 } = {x∈Z | x > 0 and x < 5 }


University of Hawaii Basic Properties of Sets n  Sets are inherently unordered:

n  No matter what objects a, b, and c denote, {a, b, c} = {a, c, b} = {b, a, c} = {b, c, a} = {c, a, b} = {c, b, a}.

n  All elements are distinct (unequal); multiple listings make no difference! n  If a = b, then {a, b, c} = {a, c} = {b, c} =

{a, a, b, a, b, c, c, c, c}. n  This set contains (at most) 2 elements!


University of Hawaii Definition of Set Equality

n  Two sets are declared to be equal if and only if they contain exactly the same elements.

n  In particular, it does not matter how the set is defined or denoted.

n  Example: The set {1, 2, 3, 4} = {x | x is an integer where x > 0 and x < 5} = {x | x is a positive integer where x2 < 20}


University of Hawaii Infinite Sets n  Conceptually, sets may be infinite

(i.e., not finite, without end, unending). n  Symbols for some special infinite sets:

N = {0, 1, 2,…} the set of Natural numbers. Z = {…, –2, –1, 0, 1, 2,…} the set of Zntegers. Z+ = {1, 2, 3,…} the set of positive integers. Q = {p/q | p,q ∈ Z, and q ≠ 0} the set of Rational numbers. R = the set of “Real” numbers.

n  “Blackboard Bold” or double-struck font is also often used for these special number sets.

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University of Hawaii ICS141: Discrete Mathematics for ...janst/141/lecture/07-Proofs.pdfDefinition: An integer n is called odd iff n=2k+1 for some integer k; n is even iff n=2k for